How to show that the set of all non-decreasing functions $f:\mathbb{Z} \rightarrow \{ 0, 1\}$ is countable?

Question

Given the set $A = \{f:\mathbb{Z} \rightarrow \{ 0,1 \} |$ if $n \geq m$ then $f(n) \geq f(m) \}$

How would I go about showing $A$ is countable?

Thank you!

Answer 1

Hint If $f$ is not constant, show that there exists a smallest $k$ such that $f(k)=1$. Here smallest means $\forall m< k$ we have $f(m)\neq 1$.

What can you say about $f$ then?

Answer 2

We can draw the conclusion:

If the function isn't always $0$ and isn't always $1$, then it must be $0$ in the left and $1$ in the right. The point where the function changes its value has countable number of choices.